The calcs needed is the heat dissapated by the regulator which is: heat (Watts) = (Vin - Vout) * current (Amps).For the LM317 you need to calculate the resistor but the info for this is in the data sheet.

How much current does your 9V adapter output?How much current does your circuit require?

Assuming you actually have a 9V AC output, and you want a 6V AC supply, then you should use a power transformer with a 3:2 winding ratio. Supply 9V AC to the "3" side, receive 6V AC on the "2" side. Magic! :-)

Btw: Some people call anything that plugs into the wall an "AC" adapter, because it adapts the "AC" from the wall, even if it generates DC. If the supply generates 9V DC, and you want 6V DC, then the best way to do that is with a switching DC DC converter (step-down converter, typically "buck" converter.) Pololu sells some decent ones. Hobby stores also sell them as "UBECs."

dc to ac then back to dc makes little sense. you need to clearly describe your goals and applications to make more sense and get better help.

also note that common linear regulators like 7806 (you can make one with a 7805 and diode) will not supply anything close to 1a w/o a heat sink. a BIG one. even 210ma is borderline w/o one. it will run hot. you need to use a dc-dc converter. assuming there is a real application here which is not clear.

Beginner question are confusing.I took it to mean an AC to DC adapter (wall wart) with 9V output (DC). These are very common so went with that. 9VAC output adapters are less common these days.Guess I should have had the OP explain exactly what he/she has first.

If you are talking about an AC adapter that outputs AC voltage, then you have to use a transformer; a regulator will not work.If you are talking about a wall wart that actually outputs DC voltage, then you have to use a regulator, either linear (7806, LM350, etc) or switching (something from http://www.pololu.com/catalog/category/131 perhaps.)

Transformers are used to change the voltage of an alternating current signal.Regulators are used to change te voltage of a direct current signal.You need to know which one it is you're dealing with.

So, if you have a 9 volt DC supply, all you need is a regulator (a capacitor on the input and output are often a good idea also). If you have a 9 volt AC supply, you need a rectifier (typically a bridge rectifier), some filtering and a regulator.